Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. DOI 10.1007/s40010-015-0239-8
RESEARCH ARTICLE
Optimal Approximate Solution for a-Multiplicative Proximal Contraction Mappings in Multiplicative Metric Spaces Chirasak Mongkolkeha1 • Wutiphol Sintunavarat2
Received: 21 August 2014 / Revised: 31 July 2015 / Accepted: 18 September 2015 Ó The National Academy of Sciences, India 2015
Abstract In this work, we introduce the new concept of a-multiplicative proximal contraction mappings in the setting of multiplicative metric spaces and we also solve optimal approximate solution for such mappings. Our main results improve and extend several results in the literature. We also state some illustrative examples to claim that our results properly generalizes some results in literature. Keywords Best proximity points Multiplicative metric spaces a-Multiplicative proximal contraction mappings a-Proximal admissible mappings Mathematics Subject Classification
47H09 47H10
the concept of multiplicative Banach’s contraction mapping and proved fixed point results for such mapping in multiplicative metric spaces. In this paper, we introduce the new class of proximal contractions in the framework of multiplicative metric spaces which is more general than classes of proximal contraction mappings in [3] and multiplicative Banach’s contraction for non-self mapping. We also give the necessary condition to solving the optimal approximate solution for such mappings and give the example of a nonlinear mapping which is not applied by the results in literature but can be applied to our results. Our main results generalize, extend and improve the corresponding results on the topics given in the literature.
1 Introduction
2 Preliminaries
Bashirov et al. [1] studied the multiplicative calculus and defined a new distance so called multiplicative distance. By ¨ zavs¸ ar and C using this idea, O ¸ evikel [2] introduced the concept of multiplicative metric spaces and studied some topological properties in such space. They also introduced
In this section, we give some definitions and basic concept of multiplicative metric space for our consideration. Throughout this paper, we denote by N, Rþ and R the sets of positive integers, positive real numbers and real numbers, respectively.
& Wutiphol Sintunavarat
[email protected];
[email protected] Chirasak Mongkolkeha
[email protected] 1
Department of Mathematics, Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng-Saen Campus, Nakhonpathom 73140, Thailand
2
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand
Definition 2.1 ([1]) Let X be a nonempty set and a mapping d : X X ! R satisfying the following conditions: (M1 ) (M2 ) (M3 )
dðx; yÞ 1 for all x; y 2 X and dðx; yÞ ¼ 1 if and only if x ¼ y, dðx; yÞ ¼ dðy; xÞ for all x; y 2 X, dðx; zÞ dðx; yÞ dðy; zÞ for all x; y; z 2 X (multiplicative triangle inequality).
Then d is called a multiplicative metric on X and the ordered pair (X, d) is called multiplicative metric space.
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Example 2.1 ([2]) Let j j : Rþ ! Rþ defined by 8 < a if a 1; jaj ¼ 1 : if a\1 a and a mapping d : ðRþ Þn ðRþ Þn ! R defined by x1 x2 xn d ðx; yÞ ¼ ; y1 y2 yn where x ¼ ðx1 ; x2 ; . . .; xn Þ; y ¼ ðy1 ; y2 ; . . .; yn Þ 2 ðRþ Þn . Then ðX; d Þ is a multiplicative metric space. Definition 2.2 ([2]) Let (X, d) be a multiplicative metric space, x 2 X and e [ 1. 1.
The multiplicative open ball of radius e with center x is denoted by Be ðxÞ and defined by
Be ðxÞ :¼ fy 2 X : dðx; yÞ\eg: 2.
The multiplicative closed ball of radius e with center x is denoted by Be ðxÞ and defined by
Be ðxÞ :¼ fy 2 X : dðx; yÞ eg: Definition 2.3 ([2]) Let (X, d) be a multiplicative metric space. The sequence fxn g in X is said to be multiplicative convergent to the point x 2 X, which is denoted by xn ! x as n ! 1, if for any multiplicative open ball Be ðxÞ, there exists N 2 N such that xn 2 Be ðxÞ for all n N. Definition 2.4 ([2]) Let (X, d) be a multiplicative metric space and fxn g be a sequence in X. The sequence fxn g is called a multiplicative Cauchy sequence if, for all e [ 1, there exists N 2 N such that dðxm ; xn Þ\e for all m; n N. Theorem 2.1 ([2]) Let (X, d) be a multiplicative metric space and fxn g be a sequence in X and x 2 X. Then the following assertions hold: 1. 2. 3.
xn ! x as n ! 1 if and only if dðxn ; xÞ ! 1 as n ! 1. If the sequence fxn g is multiplicative convergent, then the multiplicative limit point is unique. fxn g is a multiplicative Cauchy sequence if and only if dðxn ; xm Þ ! 1 as m; n ! 1.
Theorem 2.2 ([2]) Let (X, d) be a multiplicative metric space and fxn g and fyn g be two sequences in X and x; y 2 X. If xn ! x and yn ! y as n ! 1, then dðxn ; yn Þ ! ðx; yÞ as n ! 1. Definition 2.5 ([2]) Let (X, d) be a multiplicative metric space and A X. A point x 2 A is said to be a multiplicative interior point of A if there exists an e [ 1 such that Be ðxÞ A. The collection of all interior points of A is called multiplicative interior of A and denoted by int(A).
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Definition 2.6 space. 1.
2.
([2]) Let (X, d) be a multiplicative metric
A subset A X is called a multiplicative open in (X, d) if every point of A is a multiplicative interior point of A, i.e., A ¼ intðAÞ. A subset S X is called multiplicative closed in (X, d) if S contains all of its multiplicative limit points.
Theorem 2.3 ([2]) Let (X, d) be a multiplicative metric space and S X. The following assertions are equivalence. 1. 2. 3.
A subset S is multiplicative closed. The complement of S is multiplicative open. Every multiplicative convergent sequence in S has a multiplicative limit point that belongs to S.
Theorem 2.4 ([2]) Let (X, d) be a multiplicative metric space and S X. Then (S, d) is complete if and only if S is multiplicative closed. Theorem 2.5 ([2]) Let ðX; dX Þ and ðY; dY ) be two multiplicative metric spaces, f : X ! Y be a mapping and fxn g be any sequence in X. Then f is multiplicative continuous at the point x 2 X if and only if f ðxn Þ ! f ðxÞ as n ! 1 for every sequence fxn g with xn ! x as n ! 1. Here, we give the notations A0 , B0 and d(A, B) for nonempty subsets A and B of a multiplicative metric space (X, d) in the same sense in metric spaces. Let A and B be nonempty subsets of a multiplicative metric space (X, d), we recall the following notations and notions that will be used in what follows. dðA; BÞ :¼ inffdðx; yÞ : x 2 A and y 2 Bg; A0 :¼ fx 2 A : dðx; yÞ ¼ dðA; BÞ for some y 2 Bg; B0 :¼ fy 2 B : dðx; yÞ ¼ dðA; BÞ for some x 2 Ag: Definition 2.7 Let A be a nonempty subset of a multiplicative metric space (X, d). A mapping g : A ! A is said to be isometry if dðgx; gyÞ ¼ dðx; yÞ for all x; y 2 A. Definition 2.8 Let A and B be nonempty subsets of a multiplicative metric space (X, d). A point x 2 A is called a best proximity point of a non-self mapping T : A ! B if dðx; TxÞ ¼ dðA; BÞ. Remark 2.1 We note that a best proximity reduces to a fixed point if the underlying mapping is a selfmapping. Definition 2.9 A subset A of a multiplicative metric space (X, d) is said to be approximatively compact with respect to B X if every sequence fxn g in A with dðy; xn Þ ! dðy; AÞ for some y 2 B has a convergent subsequence.
Optimal Approximate Solution for \alpha -Multiplicative Proximal Contraction Mappings inldots
Remark 2.2 It is easy to see that each set is approximatively compact with respect to itself. Definition 2.10 Let A and B be nonempty subsets of a multiplicative metric space (X, d) and a : A A ! ½0; 1Þ be a given mapping. A mapping T : A ! B is said to be an a-proximal admissible if it satisfies the following condition: 9 u; v; x; y 2 A; > > > = aðx; yÞ 1; ¼) aðu; vÞ 1: dðu; TxÞ ¼ dðA; BÞ; > > > ; dðv; TyÞ ¼ dðA; BÞ
3 Main Results First of all, we will introduce the new type of proximal contraction mappings in the setting of multiplicative metric spaces as follows: Definition 3.1 Let A and B be nonempty subsets of a multiplicative metric space (X, d) and a : A A ! ½0; 1Þ be a given mapping. A mapping T : A ! B is said to be an a-multiplicative proximal contraction if there exists k 2 ½0; 1Þ satisfying the following condition: 9 u; v; x; y 2 A; > = dðu; TxÞ ¼ dðA; BÞ; ¼) aðx; yÞdðu; vÞ dðx; yÞk : > ; dðv; TyÞ ¼ dðA; BÞ
there exist x0 ; x1 ; x2 2 A0 such that dðgx1 ; Tx0 Þ ¼ dðgx2 ; Tx1 Þ ¼ dðA; BÞ and aðx1 ; x0 Þ 1; (g) if fxn g is a sequence in A such that aðxnþ1 ; xn Þ 1 for all n 2 N and xn ! x 2 A as n ! 1, then aðxn ; xÞ 1 for all n 2 N. (f)
Then there exists a point x 2 A such that dðgx ; Tx Þ ¼ dðA; BÞ:
ð3:1Þ
In addition, if aða; bÞ 1 for all a; b 2 A with dðga; TaÞ ¼ dðgb; TbÞ ¼ dðA; BÞ; then x is a unique point in A such that it satisfies condition Eq. (3.1). Proof get
Starting from x0 ; x1 ; x2 2 A0 in condition (f), we
dðgx1 ; Tx0 Þ ¼ dðA; BÞ;
ð3:2Þ
dðgx2 ; Tx1 Þ ¼ dðA; BÞ;
ð3:3Þ
and aðx1 ; x0 Þ 1:
ð3:4Þ
Since T is an a-multiplicative proximal contraction mapping and g is an isometry, we get dðx2 ; x1 Þ ¼ dðgx2 ; gx1 Þ aðx1 ; x0 Þdðgx2 ; gx1 Þ
ð3:5Þ
k
dðx1 ; x0 Þ : Remark 3.1 If a : A A ! ½0; 1Þ is defined by aðx; yÞ ¼ 1 for all x; y 2 A, then an a-multiplicative proximal contraction mapping reduces to a multiplicative proximal contraction mapping due to Mongkolkeha and Sintunavarat [3]. Therefore, the concept of a-multiplicative proximal contraction self-mapping is also a generalization of Banach’s contraction mapping in the setting of multiplicative metric spaces which was introduced by ¨ zavs¸ ar and C¸evikel [2]. O Next, we give some optimal approximate solution result for a-multiplicative proximal contraction mappings in multiplicative metric spaces.
Since x2 2 A0 , TðA0 Þ B0 and A0 gðA0 Þ, there exists an element x3 2 A0 such that dðgx3 ; Tx2 Þ ¼ dðA; BÞ: From Eqs. (3.2)–(3.4) and conditions (a) and (e), It follows that aðx2 ; x1 Þ ¼ aðgx2 ; gx1 Þ 1: It follows from T is an a-multiplicative proximal contraction mapping, g is an isometry and Eq. (3.5) becomes dðx3 ; x2 Þ ¼ dðgx3 ; gx2 Þ
Theorem 3.1 Let (X, d) be a complete multiplicative metric space and A, B be nonempty closed subsets of X such that B is approximatively compact with respect to A. Suppose that a : A A ! ½0; 1Þ, T : A ! B and g : A ! A satisfy the following conditions: T is an a-multiplicative proximal contraction mapping and T is an a-proximal admissible mapping; (b) TðA0 Þ B0 ; (c) g is an isometry; (d) A0 gðA0 Þ; (e) aðgx; gyÞ ¼ aðx; yÞ for all x; y 2 A; (a)
aðx2 ; x1 Þdðgx3 ; gx2 Þ dðx2 ; x1 Þk 2
dðx1 ; x0 Þk : By the same method, we can construct the sequence fxn g in A0 such that dðgxn ; Txn1 Þ ¼ dðA; BÞ;
ð3:6Þ
aðxn ; xn1 Þ 1
ð3:7Þ
and
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dðxnþ1 ; xn Þ dðxn ; xn1 Þk
ð3:8Þ
for each n 2 N. This implies that dðxnþ1 ; xn Þ dðx1 ; x0 Þk
n
ð3:9Þ
dðgxnþ1 ; x Þ aðxn ; xÞdðgxnþ1 ; x Þ dðxn ; xÞk : This implies that lim dðgxnþ1 ; x Þ ¼ 1 and hence the n!1 sequence fgxn g converges to x . Using Theorem 2.1 (2), we obtain that gx ¼ x and then
for all n 2 N. Next, we will show that fxn g is a Cauchy sequence. Let m; n 2 N with m [ n. Then we have
dðgx; TxÞ ¼ dðx ; TxÞ ¼ dðA; BÞ:
dðxm ; xn Þ dðxm ; xm1 Þ dðxm1 ; xm2 Þ dðxnþ1 ; xn Þ
Next, to prove the uniqueness, suppose that there exists xH 2 A with x 6¼ xH and
dðx1 ; x0 Þk
m1
¼ dðx1 ; x0 Þk
m1
dðx1 ; x0 Þk
m2
dðx1 ; x0 Þk
n
dðgxH ; TxH Þ ¼ dðA; BÞ:
þkm2 þþkn
kn
dðx1 ; x0 Þ1k : This implies that dðxm ; xn Þ ! 1 as m; n ! 1, that is, fxn g is a Cauchy sequence in A. Since A is a closed subsets of complete multiplicative metric space X, then the sequence fxn g converges to some element x 2 A. For each n 2 N, we have dðgx; BÞ dðgx; Txn Þ dðgx; gxnþ1 Þ dðgxnþ1 ; Txn Þ ¼ dðgx; gxnþ1 Þ dðA; BÞ dðgx; gxnþ1 Þ dðgx; BÞ: Since, g is continuous and the sequence fxn g converges to x, then the sequence fgxn g converges to gx, that is dðgx; gxn Þ ! 1 as n ! 1. Therefore, dðgx; Txn Þ ! dðgx; BÞ as n ! 1. Since B is approximatively compact with respect to A, then there exists subsequence fTxnk g of sequence fTxn g such that converging to some element u 2 B: Further, for k 2 N, we have dðA; BÞ dðgx; uÞ dðgx; gxnk þ1 Þ dðgxnk þ1 ; Txnk Þ dðTxnk ; uÞ
From hypothesis, we obtain that aðx; xH Þ 1. Since g is an isometry and T is an a-multiplicative proximal contraction mapping, we get dðx; xH Þ ¼ dðgx; gxH Þ aðx; xH Þdðgx; gxH Þ dðx; xH Þk ; which is a contradiction. This completes the proof.
h
Taking g is the identity mapping in Theorem 3.1, we get the best proximity point results as follows: Corollary 3.1 Let (X, d) be a complete multiplicative metric space and A, B be nonempty closed subsets of X and B is approximatively compact with respect to A. Suppose that a : A A ! ½0; 1Þ and T : A ! B satisfy the following conditions: T is an a-multiplicative proximal contraction and T is an a-proximal admissible; (b) TðA0 Þ B0 ; (c) there exist x0 ; x1 ; x2 2 A0 such that dðx1 ; Tx0 Þ ¼ dðx2 ; Tx1 Þ ¼ dðA; BÞ and aðx1 ; x0 Þ 1; (d) if fxn g is a sequence in A such that aðxnþ1 ; xn Þ 1 for all n 2 N and xn ! x 2 A as n ! 1, then aðxn ; xÞ 1 for all n 2 N. (a)
Then there exists a point x 2 A such that
¼ dðgx; gxnk þ1 Þ dðA; BÞ dðTxnk ; uÞ: ð3:10Þ
dðx ; Tx Þ ¼ dðA; BÞ:
ð3:13Þ
Letting k ! 1 in Eq. (3.10), we get dðgx; uÞ ¼ dðA; BÞ and hence gx 2 A0 . From the fact that A0 gðA0 Þ, then gx ¼ gz for some z 2 A0 . By the isometry of g, we get
In addition, if aða; bÞ 1 for all a; b 2 A with
dðx; zÞ ¼ dðgx; gzÞ ¼ 1
then x is a unique point in A such that it satisfies condition Eq. (3.13).
and thus x ¼ z 2 A0 . Since TðA0 Þ B0 , we get dðx ; TxÞ ¼ dðA; BÞ
ð3:11Þ
for some x 2 A. Inequality Eq. (3.7) and condition (g) yield that aðxn ; xÞ 1
ð3:12Þ
for all n 2 N. From Eqs. (3.6), (3.11) and the amultiplicative proximal contractive condition of T, we have
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dða; TaÞ ¼ dðb; TbÞ ¼ dðA; BÞ;
Taking aðx; yÞ ¼ 1 for all x; y 2 A in Theorem 3.1, we obtain the following result. Corollary 3.2 ([3]) Let (X, d) be a complete multiplicative metric space and A, B be nonempty closed subsets of X such that A0 and B0 are nonempty and B is approximatively compact with respect to A. Suppose that T : A ! B and g : A ! A satisfy the following conditions:
Optimal Approximate Solution for \alpha -Multiplicative Proximal Contraction Mappings inldots
(a) T is a multiplicative proximal contraction; (b) TðA0 Þ B0 ; (c) g is an isometry; (d) A0 gðA0 Þ.
dðð0; uÞ; ð0; vÞÞ ¼ dðð0; 4Þ; ð0; 9ÞÞ ¼ e5 e ¼ dðð0; 2Þ; ð0; 3ÞÞ ¼ dðð0; xÞ; ð0; yÞÞ
Then there exists a unique point x 2 A such that dðgx ; Tx Þ ¼ dðA; BÞ:
[ dðð0; xÞ; ð0; yÞÞk
Moreover, for any fixed x0 2 A0 , the sequence fxn g defined by dðgxn ; Txn1 Þ ¼ dðA; BÞ converges to the element x . Taking A ¼ B ¼ X and g is the identity mapping in Corollary 3.2, we get classical following results: Corollary 3.3 ([2]) Let (X, d) be a complete multiplicative metric space and T : X ! X be a multiplicative Banach contraction, then T has a unique fixed point.
for all k 2 ½0; 1Þ. Therefore, T does not satisfies multiplicative proximal contractive condition. Hence, the results of Mongkolkeha and Sintunavarat [3] can not use for this case. Here, we will show that Theorem 3.1 can be used for this case. Now we define a mapping a : A A ! ½0; 1Þ by 1; x; y 2 ½1; 1 ; aðð0; xÞ; ð0; yÞÞ ¼ 0; otherwise for all ð0; xÞ; ð0; yÞ 2 A. (a)
Next, we give an example to illustrate Theorem 3.1. Example 3.1 X ! R by
Let X ¼ R2 . Define the mapping d : X
dððx1 ; x2 Þ; ðy1 ; y2 ÞÞ ¼ ejx1 y1 jþjx2 y2 j for all ðx1 ; x2 Þ; ðy1 ; y2 Þ 2 X. Then (X, d) is a complete multiplicative metric space. Let A ¼ fð0; xÞ : x 2 Rg
and
B ¼ fð1; yÞ : y 2 Rg:
It is easy to see that dðA; BÞ ¼ e, A0 ¼ A, B0 ¼ B and B is approximatively compact with respect to A. Define mappings T : A ! B and g : A ! A by 8 < ð1; xÞ; x 2 ½1; 1 2 Tðð0; xÞÞ ¼ : ð1; x2 Þ; otherwise ; and gðð0; xÞÞ ¼ ð0; xÞ for all ð0; xÞ 2 A. Now we will show that T is not a multiplicative proximal contraction mapping. For ð0; uÞ ¼ ð0; 4Þ; ð0; vÞ ¼ ð0; 9Þ; ð0; xÞ ¼ ð0; 2Þ and ð0; yÞ ¼ ð0; 3Þ, we get dðð0; uÞ; Tðð0; xÞÞÞ ¼ dðð0; 4Þ; ð1; 4ÞÞ ¼ e ¼ dðA; BÞ and dðð0; vÞ; Tðð0; yÞÞÞ ¼ dðð0; 9Þ; ð1; 9ÞÞ ¼ e ¼ dðA; BÞ but
We will show that T is an a-multiplicative proximal contraction with k ¼ 12. Let ð0; uÞ; ð0; vÞ; ð0; xÞ; ð0; yÞ 2 A such that x; y 2 ½1; 1 and
dðð0; uÞ; Tðð0; xÞÞÞ ¼ dðA; BÞ ¼ e and dðð0; vÞ; Tðð0; yÞÞÞ ¼ dðA; BÞ ¼ e: Then we obtain that u ¼ 2x and v ¼ 2y. This implies that x y aðð0; xÞ; ð0; yÞÞdðð0; uÞ; ð0; vÞÞ ¼ d 0; ; 0; 2 2 1
¼ e2jxyj 1 ¼ dðð0; xÞ; ð0; yÞÞ 2 : In otherwise, we get a-multiplicative proximal contractive condition holds. Therefore, T is an a-multiplicative proximal contraction with k ¼ 12. Also, it is easy to see that T is an a-proximal admissible mapping. (b) It is easy to see that TðA0 Þ B0 . (c) For each ð0; xÞ; ð0; yÞ 2 A, we get dðgð0; xÞ; gð0; yÞÞ ¼ dðð0; xÞ; ð0; yÞÞ ¼ ejxyj ¼ dðð0; xÞ; ð0; yÞÞ: This implies that g is an isometry. (d) We can see that A0 gðA0 Þ. (e) For each ð0; xÞ; ð0; yÞ 2 A, we get aðgð0; xÞ; gð0; yÞÞ ¼ aðð0; xÞ; ð0; yÞÞ ¼ aðð0; xÞ; ð0; yÞÞ: (f)
It is easy to see that there exist x0 ; x1 ; x2 2 A0 such that dðgx1 ; Tx0 Þ ¼ dðgx2 ; Tx1 Þ ¼ dðA; BÞ and aðx1 ; x0 Þ 1.
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(g)
Let fxn g be a sequence in A such that aðxnþ1 ; xn Þ 1 for all n 2 N and xn ! x 2 A as n ! 1. Then we get
by Faculty of Science and Technology, Thammasat University under the TU Research Scholar, Contract No. 11/2558.
xn ; x 2 fð0; aÞ : a 2 ½1; 1 g: This implies that aðxn ; xÞ 1 for all n 2 N. Now all hypotheses in Theorem 3.1 hold and so there exists a unique point x in A such that dðgx ; Tx Þ ¼ dðA; BÞ. In this case, x ¼ ð0; 0Þ 2 A is an element such that dðgx ; Tx Þ ¼ dðA; BÞ. Acknowledgments C.M. would like to thank Kasetsart University Research and Development Institute (KURDI) for financial support in this work. W.S. gratefully acknowledge the financial support provided
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References ¨ zyapici A (2008) Multiplicative 1. Bashirov AE, Kurpinar EM, O calculus and its applications. J Math Anal Appl 337:36–48 ¨ zavs¸ ar M, C¸evikel AC (2012) Fixed points of multiplicative 2. O contraction mappings on multiplicative metric spaces, Math GN: 23 May 2012 3. Mongkolkeha C, Sintunavarat W (2015) Best proximity points for multiplicative proximal contraction mapping on multiplicative metric spaces. J Nonlinear Sci Appl 8:1134–1140
